Irreducible polynomial - Imported Knowlpedia Content

Let $R$ be an integral domain. A polynomial $f\in R[x]$ is irreducible if $f$ is nonzero, not a unit, has positive degree, and whenever $f=gh$ with $g,h\in R[x]$ , then either $g$ or $h$ is a unit.

Remarks

Over a field, irreducible polynomials generate maximal ideals in $R[x]$ and define finite field extensions. Practical tests include Eisenstein’s criterion, and primitive hypotheses are often used to compare factorization in $R[x]$ and in $\mathrm{Frac}(R)[x]$ .

Examples

In $\mathbb{R}[x]$ , the polynomial $x^2+1$ is irreducible.
In $\mathbb{Q}[x]$ , $x^3-2$ is irreducible (Eisenstein at $2$ ).
In any $R[x]$ , $x^2-1=(x-1)(x+1)$ is reducible.